Abstract
In this paper we propose and compare two methods to optimize the numerical computations for the diffusion term in a nonlocal formulation for a reaction-diffusion equation. The diffusion term is particularly computationally intensive due to the integral formulation, and thus finding a better way of computing its numerical approximation could be of interest, given that the numerical analysis usually takes place on large input domains having more than one dimension. After introducing the general reaction-diffusion model, we discuss a numerical approximation scheme for the diffusion term, based on a finite difference method. In the next sections we propose two algorithms to solve the numerical approximation scheme, focusing on finding a way to improve the time performance. While the first algorithm (sequential) is used as a baseline for performance measurement, the second algorithm (parallel) is implemented using two different memory-sharing parallelization technologies: Open Multi-Processing (OpenMP) and CUDA. All the results were obtained by using the model in image processing applications such as image restoration and segmentation.
Highlights
Image processing tasks such as noise filtering, image deblurring and shape extraction are often based on partial differential equation (PDE) models; see, e.g., [1,2,3].After Perona and Malik introduced an anisotropic diffusion model for image denoising [2], their PDE-based model became a the starting point for numerous other derived models [3] which aim to improve it
OpenMP (Open Multi-Processing) is an application program interface implemented by most of the modern C/C++ compilers which allows for fast development of portable multi-threaded applications based on shared memory
We introduced a discretization method for a nonlocal reaction-diffusion model applied to a 2D input domain
Summary
Image processing tasks such as noise filtering, image deblurring and shape extraction (image segmentation) are often based on partial differential equation (PDE) models; see, e.g., [1,2,3]. The image being analyzed forms the input domain, Ω, and time is a synthetic variable used to describe the iterative process; x, y ∈ Ω, ys ∈ ∂Ω and t ∈ (0, T ]; v(t, x ) is the unknown function used to model the evolution of the image in time (after each processing step). Applications to image restoration were well studied in [1], wherein a local form of a newly introduced anisotropic diffusion system was used. This model can be seen applied to image processing and other domains in [2,3,5,6]. From the computational point of view, the ND term will most often lead to intense computational requirements given the two integrals that need to be evaluated and that usually, in practical applications, Ω is a large domain
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